An inequality uses one of the signs <, >, ≤, ≥ instead of an equals sign to compare two quantities.
The equation x = 3 has a single solution, but the inequality x > 3 is satisfied by every number greater than 3, so its solution is a whole stretch of the number line.
A ride you can take only if you are at least 120 cm tall, or how many items fit into a budget of ten dollars, are everyday conditions that are really inequalities.
Solving an inequality works almost like solving an equation; the one difference is that the sign turns around when both sides are multiplied or divided by a negative number.
An inequality sign instead of an equals sign
👀 Inequalities and their solutions
①Only one number, 3, makes the equation x = 3 true.
②The inequality x > 3 is made true by 3.1, by 4, by 100: its solution is not one number but a range.
③On the number line, < and > leave out the boundary and are drawn with an open circle (○); ≤ and ≥ include it and are drawn with a filled circle (●).
④In words, ≥ is "at least", ≤ is "at most", > is "more than" and < is "less than".
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The solution of an inequality shaded on the number line, set by the boundary value and the sign
3
>
Multiplying by a negative number flips the sign
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Where the two points of 2 < 5 move on the number line when both sides are multiplied by the same number
2
Property of inequalities
If a < b and c < 0, then ac > bc and a/c > b/c
Adding or subtracting the same number, or multiplying or dividing by a positive number, keeps the direction of the sign. Only multiplying or dividing by a negative number reverses it.
Solving linear inequalities
Order of steps
expand brackets → move terms → collect like terms → divide by the coefficient of x
The order is the same as for a linear equation. If the coefficient you divide by at the end is negative, reverse the sign.
📝 One worked solution
①In 2x − 3 > 5, moving −3 to the right gives 2x > 8.
②Dividing both sides by the positive number 2 gives x > 4, and the sign stays the same.
③On the number line, draw an open circle at 4 and an arrow to the right. 4 itself is not a solution.
Using inequalities
💡 Turning a sentence into an inequality
①Let x be the quantity you want to find.
②Choose the sign from words such as "at least", "at most", "more than", "less than" or "does not exceed", and write the inequality.
③After solving, keep only the values that fit the conditions of the problem. If the answer must be a whole number, such as a count of items, only the natural numbers inside the range count.
Worked examples
Example 1
Solve the inequality 2x − 3 > 5.
1
Move the constant −3 to the right-hand side.
2x > 5 + 3 = 8
2
Divide both sides by the positive number 2; the sign stays the same.
x > 4
▸ x > 4
Dividing by a positive number never changes the direction of the sign.
Example 2
Solve the inequality −3x + 1 ≥ 7.
1
Move the constant 1 to the right-hand side.
−3x ≥ 6
2
Dividing both sides by the negative number −3 reverses the sign.
x ≤ −2
▸ x ≤ −2
Check with x = −3: −3·(−3) + 1 = 10 ≥ 7 is true, while x = 0 gives 1 ≥ 7, which is false. The reversed answer is right.
Exam checklist
Solution of ax + b > 0
x > −b/a if a > 0, x < −b/a if a < 0
The same expression gives solutions in opposite directions depending on the sign of the coefficient a.
Grade-8 school exam type
How many natural numbers x satisfy 5 − 2x ≥ −1?
①1
②2
③3
④4
⑤5
▸ ③ 3
1
Move 5 to the right-hand side; dividing by the negative number −2 reverses the sign.
−2x ≥ −6 ⇒ x ≤ 3
2
Count the natural numbers that satisfy x ≤ 3.
x = 1, 2, 3 ⇒ 3 numbers
🎯 Exam points
①On the number line, < and > use an open circle (○); ≤ and ≥ use a filled circle (●).
②Multiplying or dividing both sides by a negative number reverses the sign; this is the most common mistake.
③When a term moves to the other side, its sign changes.
④"At least" and "at most" include the boundary; "more than" and "less than" do not.
⑤In word problems that require natural numbers, count only the natural numbers in the range.